According to this site and the Honor FAQ which I got most of my information from.
There are four types of honor:
Helpful - There are those summoners who have the know-how to help you step up your game and are always willing to share. If one of your teammates makes you feel like you’re attending a master class on League of Legends, be sure to recognize him for being a helpful teammate.
Friendly - Ever meet one of those summoners who – win or lose – is just a pleasure to play with? This acknowledgement goes out to those unsung heroes of the Fields of Justice whose friendly demeanor keeps everyone having a good time even when the chips are down.
Teamwork - Some players really put the needs of the team over their personal interests. These players lead the way on the Fields of Justice, support struggling teammates, and are always the first to step up and select a role the team needs. This acknowledgement is for all those players who understand that a chain is only as strong as its weakest link.
Honorable Opponent - Sometimes you run into an opponent who is humble in victory and graceful in defeat. Be sure to acknowledge those adversaries who embody the spirit of sportsmanship as Honorable Opponents.
You currently get some profile indicators, as well as other bonuses.
The profile indicator: (Icons left to right are: Friendly , Helpful , Teamwork , Honorable Opponent)
First bonus for getting that you are a great teammate is this during champion select:
Same thing but for being an honorable opponent
These also show up during the loading screen
The popup once you are alerted that you have received one of these alerts
Right now the exact amount of times you need to be honored is unknown. This is more then likely due to the honor votes being weighted in the background. Someone who honors just their friends votes count less then random people you play in matchmaking.
Right now the only way to loose honor is if you are reported to the Tribunal and found guilty of of a infraction. This resets all your honor to zero.